Efficient two-scale simulations of engineering structures using the Hashin-Shtrikman type finite element method

An efficient FE2-like numerical homogenization approach for nonlinear microstructures is proposed using the Hashin-Shtrikman type finite element (HSFE) method to solve the microscale problem. The latter combines the small computational effort of Hashin-Shtrikman type homogenization approaches with the accuracy of full-field FE-solutions. The key point is a reduced order method based on a Hashin-Shtrikman type variational formulation combined with data-clustering, which is based on offline FE-simulations of microstructures (snapshots). The presented microscopic model has significantly less microscopic degrees of freedom in comparison to the classic FE2-method. The number of stress computations within the microstructure is highly reduced. The tangent operator, which incorporates the coupling between the microscopic and macroscopic scale, is derived analytically. Different numerical examples are investigated, where a nonlinear RVE is attached to each integration point of a macrostructure. A comparison to full-field FE-simulations shows that the macro-response and local fields are well captured by the HSFE method.